When a free proton captures a free electron

If we shoot a single low energy proton directly into the path of a single low energy electron, both being truly free in space, what theory describes the change in the spatial distribution of the free electron as it changes from a "focused" point-like particle into a "cloud-like" point particle that envelops the proton?

Proton and electron are anti-particles of each other . The result may end up being complete annihilation with energy being radiated in terms of radiation.
BJ

The anti-particle equivalent of the electron is the positron, the anti-particle equivalent of the proton is the anti-proton. A fundamental particle such as an electron cannot not have a composite particle as its anti-particle even though the charge is different.

Since the electron loses energy by being in a lower energy level around the proton, a photon has to be emmited as it falls from it's initial energy to the new energy. Even if you assumed the electron had no kinetic energy in the unbound state, it gains energy falling into the electric potential well.

Since the electron loses energy by being in a lower energy level around the proton, a photon has to be emmited as it falls from it's initial energy to the new energy. Even if you assumed the electron had no kinetic energy in the unbound state, it gains energy falling into the electric potential well.

I'm puzzled. When I read HyperPhysics about Electron Capture I am reading about Radioactivity wherein (7)Be captures and electron and converts into (7)Li with a photon emitted.

That is a different kind of capture. You are converting one of the protons into a neutron. I'm not sure if "capture" is even the most correct name for what we are talking about. Electron recombination? I don't know.

If by "capture" you mean the electron joins the proton in a new Hydrogen atom, that can certainly happen. But it only happens at extremely low energies. Just a handful of eV, no more. And the energies and directions of the two particles need to be just right.

But if you're talking about some kind of absorbsion going on, things are different.

Let's say you shoot an electron directly at a proton, so that they collide. They do this at the HERA lab. It's not going to be like two billiard balls colliding, because the proton is really made of smaller things. At sufficient energies, the electron is going to collide with one of the quarks or gluons that the proton is made of, if it collides with anything at all.

There have been observations at HERA of electrons colliding with something OTHER THAN a quark or gluon. Quarks and gluons have "color" attributes, but sometimes you see an electron interacting with something "colorless" within the proton. The electron appears to be "scattered" when this happens. I have no clue what's going on when that happens.

Sometimes, what happens in this kind of collision is that the proton and electron reassemble into a neutron and a neutrino. A heck of a lot of energy gets released. Sometimes, you wind up with an electron again, plus a lot of hadronic junk.

The coolest observations, however, are when an electron dives into the interior of a proton, and gets reflected straight back out the way it came. As if there was a mirror deep inside a proton. This seems to contradict the Standard Model (which would require such a reflecting object to be very very massive and also have very strong interactions with leptons), but further analysis is required before any conclusions are drawn. It may well be that the collision of an electron with a quark changes the nature of the quark so that behaves this way (some call that hypothetical creature a "lepto-quark"). Or it could be something not yet dreamt of in our philosophy.